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2:00 PM
Poincare-conjecture ? No, i thought it has been solved in every dimension ?
 
Well, that's the generalized Poincare conjecture
the original one, in $n=3$, was solved by Ricci flow
the $h$-cobordism theorem solves it in $n\ge 5$
$n=4$ is unsolved
I think it's thought to be false though
 
I always mean that: A topological manifold which is homotopy equivalent to a sphere is also homeomorphic to it, or something like that.
Or wait, you are right, its a 3-fold
 
Apparently there is a topological and a smooth version
 
https://en.wikipedia.org/wiki/Exotic_sphere

Any explicit examples in R4?
 
2:08 PM
That's the question @Secret
There are exotic $S^7$s
 
I dont know whether the conjecture is wrong in the case n=4
 
and the fact that there are an infinite number of exotic $\Bbb R^4$s makes people think there are exotic $S^4$s (I think)
 
exotic spheres only appear in theories with dimensions higher than 7, I think.
 
23
Q: Exotic differentiable structures on R^4?

jeremyThis was going to be a comment to Differentiable structures on R^3, but I thought it would be better asked as a separate question. So, it's mentioned in the previous question that $\mathbb{R}^4$ has uncountably many (smooth) differentiable structures. This is a claim I've certainly heard before...

What is an explicit example of this in terms of set theory or some other notation?
So far I keep finding they say there is an example, but they have not say what it is
 
Jim
2:27 PM
just rejected this suggested edit
now this edit is suggested
same edit by same user. Apparently one reject is not enough of a hint
 
why did they capitalize stuff
@Jim dw, voted in your favor
 
Jim
the OP? because it's a title and they felt capitalizing titles is standard I guess
 
is one supposed to capitalize email subject lines?
 
Jim
depends on context. Informal, who cares. Formal, at least I do
 
This guy seems to have ~200 rep from questions & answers if I estimate correctly ... that must be a pretty impressive editing history
 
2:33 PM
@Jim Did you ever encounter almost complex structures in your studies?
 
Jim
@0celo7 I don't think so. I'm not good with terminology, but I'm going to lean on the side of "no"
 
I need ACM
 
Jim
yes, he's much more math-focused
 
I've been tricked into reading an actual PhD level book
 
Jim
you can just call it graduate level. In terms of content or material, there isn't much actual difference between a masters and a PhD. Masters degrees cover most if not all of the actual coursework/learning in graduate school, the distinguishing feature for a PhD is mostly the added work they put into it: publications, etc. Obviously that may vary from institution to institution, but it's the general trend I've noticed
 
2:40 PM
(where on earth is ACM, I have not seen him for ages)
 
Jim
That said, an extra 4 years of research does give one the experience to dive into unfamiliar physics a bit more easily
 
@Jim you're talking about the US system?
 
Jim
@Sanya no, I'm Canadian
 
@Jim The intro literally says it's a PhD course
 
Jim
I assume they are mostly the same
 
2:42 PM
It's written by a German
 
Jim
@0celo7 wtf is a PhD course? Can you only take it if studying a PhD? This raises so many questions.
 
I dunno
 
@yuggib How does the notion of "individual subsystems" disappear as I progressively made two qubits more entangled from 0 to 100%., because from the density matrix of the full system, all I see is the diagonal entries of the matrix goes from some fractions that adds to 1 to becoming 1/2 for both, and that when > 0% entangled, I cannot split them into subsystems via tensor products?
 
@Jim yeah, without wanting to hurt your feelings, from my side of the atlantic it appeared like the edutcation system is pretty similar even if I'm aware that it's not the same country - just interesting to hear that there's no big difference between Master & PhD
 
Jim
Is there some secret PhD society where they have hidden physics that nobody speaks about around non-PhD people. And when studying for a PhD, you take courses where they are like "Okay, here is the REAL science that we keep secret because everyone else is stupid"
 
2:45 PM
This isn't science
 
Jim
@Sanya I'd say 4 years of dedicated original research is a pretty substantial difference
 
maybe in the U.S. they usually don't have master program for physics, so PhD courses just mean graduate courses
 
@Jim that's what I would have expected - so maybe I just misread you
 
or more precise: the PhD in Physics in the u.s. usually include master program. (to my knowledge)
 
Holy moley this problem
 
Jim
2:48 PM
@Sanya Coursework learning, there's little difference. That means, except in the field of one's research, a masters and a PhD have about the same level of understanding of physics in general. But in the field of research, masters grad <<< PhD grad
 
How do you calculate the differential of a flow
 
@Jim ah ok ... what is coursework during a PhD?
 
Jim
@Sanya some institutions don't have PhD courses. They do all the learning in masters. This is variable, however. But usually, a masters student can take all the same courses as a PhD student
just with less time to do it all
 
@Jim thanks for the explanation :) that's a nice system ...
 
3:09 PM
@Secret at Wacken I think.
Wacken Open Air (W:O:A) is a summer open-air heavy metal music festival. It takes place annually in the small village of Wacken in Schleswig-Holstein, northern Germany. With 80,000 festival visitors, and including personnel a total of roughly 86,000 attendees in 2011, it attracts various kinds of metal and hard rock music fans. The festival was first held in 1990 as a small event for local German bands. W:O:A is usually held at the beginning of August and lasts now four days. It is currently considered the biggest heavy metal festival in the world. The festival ends traditionally on the first Sunday...
 
3:19 PM
what is so hard to understand about full reference ... sometimes I just don't understand people ... ^^"
 
o yes, forgot he's a metal fan
 
"Remarkably, all 75,000 tickets were sold out within 43 hours in 2014, 12 hours in 2015 and 23 hours in 2016," they are so crazy about this XD
 
@Secret It ended on Saturday, but personal experience suggests these things take some time to recover from :-)
Should I answer this, or is it as Gary suggests a lazy question?
0
Q: Calculating spontaneous decay rate?

AdamIs it possible to calculate the spontaneous decay rate of an atom's (for sake of "simplicity" hydrogen) excited state in qft, if so how and if not why not?

 
it basically depends on your learning; most of the questions here are lazy questions in a sense that given a good library, research skills and time they are absolutely doable by yourself - do you want to answer it? then go ahead :D
 
3:34 PM
@JohnRennie what if he's not
what if he's lying to get away from us
 
Look, he's a long haired, German, Goth Metal fan. Where else would he be? :-)
 
in an average german physics class, you see at least one person with a wacken t-shirt, usually more ;) so the chance is high
 
Jim
This is the way the Jim ends; not with a bang but with tedious mathematical calculations. — Jim 6 mins ago
^ got nothing for that... no response whatsoever
 
what
@JohnRennie you don't want me to answer that
 
@0celo7 :-)
 
4:08 PM
@JohnRennie it rhyme with BDSM club
Whoops
Dat Rick Ross self-rhyme
If you don't get the reference I can hook u up @JohnRennie
He has one song where he rhymes the same word with itself 10 times
 
Rhymes with BDSM club? I'm far too old and boring to get that reference.
 
Like really
You don't know what a BDSM club is?
 
I know what a BDSM club is (not that I've ever been to one officer) but I don't get the reference to rhyming with BDSM club
 
Oh
It's a shitty Rick Ross reference
My favorite rapper but he likes rhyming words with themselves
Definitely not a top lyricist
 
Anyway why would ACM go to a BSDM club? All he has to do is answer questions here and people start flagelating him.
 
4:13 PM
Not the same as a real German whip if you know what I mean
 
German whip = car doesn't it?
says John proving he's not completely out of touch.
 
I wasn't talking about it in the rap sense.
I'm saying he likes whipping people
 
BTW did you listen to any George the Poet, or is English rap just too awful to contemplate?
Though I wouldn't say George the Poet is really rap.
 
I listened to the song
 
And ... ?
 
4:17 PM
If you want something mainstream with a similar (?) message, there's She wish she was a nigga
I'm not sure if it's horribly sexist or valid
That's the problem with gangster rap sometimes
It was more like slam poetry than rap
 
Good grief - a new low point :-(
-2
A: How come a rigid body has 6 degrees of freedoms (DOFs) ? Isn't velocity a DOF?

user126711 i want complete answers for all qutions step by step dint copy internet

 
and not to mention, the user's not even an answer
Hmm, looks like if we are stuck with some series expansion, we can consult help from lego to solve it
 
4:44 PM
Hola
 
I didnt know ACM was a metal fan
 
5:30 PM
No messages ?
 
5:45 PM
@WilliamBulmer Hola
 
It's so quiet here
 
@Jim You should not have rejected that.
 
Well, I met with a physics post doc last week to share a set of original research I have been working on. I was surprised to find that he thinks I shouldn't try for a Ph.D. and should, instead collate my ideas into a 4-10 page research paper.
 
@WilliamBulmer That's.... odd.
@WilliamBulmer Monday morning :)
People are at work etc.
 
he thinks that obtaining a PhD would be impractical because the theoretical stuff I am interested in is not currently "in vogue," and that I would be forced to do research in things I don't care about, and also that the skills I would gain doing a PhD aren't worth the effort
Yeah, I know. I should really be working, myself
I come here mostly to lurk, and absorb physics talk from people who likely know better than I do
 
Jim
5:52 PM
@DanielSank the suggester added an "ing" to the title so that they could go ahead with the same edit that was already rejected. It really didn't improve the post anyway
 
Jim
@Sanya anyone who makes that many suggested edits should expect that the odd one will be rejected
 
@Jim I find it odd to be that fixed on edits anyway, but that's maybe just me ...^^
 
Jim
@Sanya well, you can't spell "Physics.StackExchange" without "odd".
 
I wouldn't be an exception here either
here as "in this respect", sorry for my lack of language skills
 
6:03 PM
@Jim The first edit did improve the post.
It brought the title into conformity with site rules.
We have a FAQ about that.
 
Jim
@DanielSank Fair enough. I'm still standing by my opinion that de-capitalizing the title is not enough to count as an actual improvement on the post. It can accompany something that actually improves the post, but I'd still reject as "no significant improvement"
 
@Jim Why?
What is gained by rejecting that edit?
It makes the questions list easier to read at no cost to yourself.
Now, instead, that user is confused about the site rules, and if they go back to that post they'll see the edit was made anyway but they got no rep for it.
 
Jim
it's not about what's gained by rejecting it. The question is what's gained by accepting it. I'd argue that de-capitalizing a title does not make it easier to read whatsoever
 
wait, the edit was to decapitalize?
 
Jim
yes
 
6:09 PM
oh shizzle, I thought it was the other way around
my b
 
@Jim Well, then you either disagree with the FAQ I linked or you just don't think that consistent style helps readability. Which is it?
 
Jim
the second one
 
@Jim Ah. Ok.
Well, consistent style does help readability.
 
Jim
FAQ is fine, that edit alone was not enough to improve readability
 
@Jim False.
:)
 
Jim
6:12 PM
the larger issue, however, was that once it was rejected, the suggester immediately suggested what was essentially the exact same edit. Ignoring the opinions of the reviewers (plural, I wasn't the only one)
 
@Jim Not surprising, give that the reviewers were acting contrary to written site policy.
 
Jim
@DanielSank seriously? I read the title perfectly fine both with and without the edit. It's true that consistency is ideal, but it should not itself be a reason to bump something to the top of the active page. Especially when it is ensuring such a minor consistency correction
 
@Jim Now you're bringing the front page list into the mix.
Editing isn't about bumping IMHO, it's about making the post better.
Who cares if it gets bumped?
 
I would appreciate it if someone could explain-to-OP/mediate/step-in/vote-to-reopen/vote-to-close here.
 
@Qmechanic I can step in.
 
Jim
6:15 PM
@DanielSank that's the whole reason edits that offer insignificant improvement should be rejected. The raison d'etre of that reject reason
 
I understand why OP wants that question in physics.
@Jim Ok, well I disagree. I think consistent title capitalization is a significant improvement.
I think style matters a lot.
So fine, continue to reject, and I'll continue to make those edits outside the review queue because I have the rep.
 
Jim
@DanielSank we can agree that style matters, we just differ in how much we think this instance of style matters
 
Never understood why high rep lets you do that, but that's the world we live in.
@Jim Indeed.
I think that it matters a lot, in particular because it helps the original author learn to use proper style.
 
Jim
I, on the other hand, think that changing the first letter of half of the words in the title to be in lower case does not improve the readability of the post enough to be worth its own edit. If they had added even one tag, I'd have approved it.
 
@Jim All you're going to do with that approach is encourage people to add crap to their edits to push them through.
Just allow surgical edits and you get higher quality content without the cruft.
Why didn't you add the tag and approve it?
 
vzn
6:22 PM
@WilliamBulmer it seems surprising to hear a Phd denigrate the achievement. what does he think is "in vogue"? what is your research on? do you have a masters? we are always looking for guest speakers in here
 
Jim
@DanielSank if the crap improves readability, I see no problem with allowing it. I feel what I more going to do is keep someone from making repeated minor edits to a question so that their answer can stay active longer and get more views. And as a robot, I feel making a judgement call about intentions in the moment is a bit beyond what should be expected of me
@DanielSank I didn't think of that at the time
 
@Jim :)
 
@vzn I thought it was odd, too. He didn't really elaborate on what was currently in vogue, so unfortunately, I can't answer that question. My research concerns Classical Electrodynamics. No, I don't have a Master's. I have 3 years formal college -training plus a few more years of private study. I am currently making my way through Rindler's GR and am trying to learn about Lie Theory, as it seems to be fairly useful to know about.
He thinks what I really need, more than anything else, is feedback on my ideas
Also, I don't want to make it sound like he endorses my ideas (yet). He just thinks it would be better for me to try to publish a paper on them, should they be sound, than for me to try obtaining a PhD
 
@WilliamBulmer That's usually very good advice.
 
6:38 PM
@DanielSank yeah, feedback would be helpful. The problem with learning on my own is that I am not around other people to tell me I have bad ideas
All anyone around me ever does when I tell them about what I know, and what I have hypothesized, is nod their head, and remark "You're a genius" or something to that effect
no matter how simplistically I try to explain things
 
What kind of work have you done?
 
@WilliamBulmer Yes.
 
It's really frustrating and isolating
 
vzn
(re icecube)
@WilliamBulmer blogs can be good for some feedback. arxiv might take your paper
 
6:42 PM
hence, I thought, I might benefit from going back to school and being in an environment where I not only can be among people who understand what I say, but learn things "the right way"
 
vzn
@WilliamBulmer it sounds like you dont have a finished undergrad degree from your sentence
 
@vzn The last classes I took were 2nd semester QM, Statistical Mechanics, and Anaytical Mechanics, and Electromagnetism
 
vzn
@WilliamBulmer (fully sympathize, but still wondering...) why stop with the undergrad work?
 
@vzn I stopped because I lost my job, which was supporting my studies
 
vzn
@WilliamBulmer sorry to hear that, yes edu is a major financial sacrifice for many esp "lately," have blogged on that some etc
 
6:47 PM
@vzn It's really frustrating. Now, to be fair, my current ob has offered to pay for my continuing studies, but only if I pull an A average
 
vzn
so what is there to research in classical electrodynamics? theoretical and/or applied?
 
@vzn Theoretical
It concerns reworking the Maxwell
 
vzn
@WilliamBulmer research/ edu is not the only thing in life that is like that :\
 
's equations in conjunction with the Poynting Vector in mathematically interestng ways
and then using that to formulate a physical hypothesis based on those new forms
Well, I should say in conjunction with the Poynting Vector and the Lorentz Force
But I risk talking about original research here
 
7:07 PM
Actually, I do have a question, which might help me. Does anyone know what the significance of the eigenvalues of principal stresses of a stress tensor is? I suppose it is likely related to treating the stress tensor as an operator, but I don't quite understand what it would be operating on. Would it be operating on field velocity?--to obtain a certain kind of field momentum density?
 
Share with friends and family
 
@0celo7 Ooh, that's cool.
No physics, but interesting, still: nature.com/news/…
 
Well, the major results for the last decade I can remember: 1) Higgs boson found 2) Gravitational waves found 3) neutrino oscillation found. Minor results: 3) QGP 4) tricky tetra- and pentaquark combinations 5) muon magnetic moment.
These don't seem very bad to me. If we compare this to the previous decades, I don't think it had been significantly better or worse, although I admit my naive impression based on remembers from the news is probably below 5 sigma. :-)
 
7:36 PM
We detected the W and Z bosons and found the accelerated expansion in the 90's
 
Well, and the WMAP + 1a supernovas + some third experiment, I forgot which... these 3 independent measurement gave an consequent and accurate answer to the age and the curvature of the Universe.
Although maybe the flat geometry was a little bit sad. Probably most of us waited a small positive curvature. Everything is spherical in the Universe, already Aristotle said, that the planets are spherical, and the sky is spherical, because the sphere is the most perfect geometrical form. :-)
We are only sad, because the LHC didn't give 42 new supersymmetric particles now. It is sad, but a desert can be maybe also attractive.
Even if it isn't really popular. :-)
 
7:56 PM
@DanielSank : Thanks.
 
@peterh : I hope the next decade will be one that captures the imagination of the public and increases their interest in physics.
 
WACKÖÖÖÖÖÖÖÖÖN
To which of the many pings in my inbox do I still need to reply? :P
 
ACM! So long I didn't see your posts, I've thought you are ignoring me! :-) You were on a vacation?
@JohnDuffield ...which would be a nice feature of the PSE... if we wouldn't expel somebody by deleting his question asking, how would neutronium look :-)
@JohnDuffield Btw, a scientific communication site and an educational site could be made possible also by euphemistic tagging, too.
 
@ACuriousMind Hello
Depends, what do the ones from me say?
I think I've solved everything except for one issue
@ACuriousMind I think you have about 5 from me saying how amazing Čech cohomology is
 
I do :D
 
8:06 PM
Oh, I have two questions
 
@peterh : noted. I think the main thing is to be nice.
 
One is less important, but perhaps more interesting to you.
 
Well, keep the technical questions for tomorrow, perhaps
I need to return my mind to business as usual first
 
The less important one involves isomorphisms of cohomology groups of augmented doubly graded differential complexes
It's technical
That sounds like random math words lol
The second one involves flows of Ricci soliton functions on the 2-sphere with almost complex Riemannian connection funness
 
vzn
@WilliamBulmer that sounds like a statics/ mechanics/ dynamics question...? where does the "stress tensor" arise, what context? it would help if you could cite something
 
8:09 PM
It's perhaps less technical
 
@JohnDuffield For example: tags could be introduced: "exercises", "educational", "conceptual"
 
@ACuriousMind Oh, and the third involves estimation of curvature scales via PDE methods, but I doubt you'll want to hear it.
 
@vzn Well, the stress tensor in question in is the Maxwell stress tensor, but I am talking, in general, about the Cauchy Stress tensor of Classical Field Theory
Specifically, the eigenvalues of the Maxwell stress tensor keep appearing in interesting ways in the work I am doing
I noticed that one of the principle stress eigenvalues is simply the EM energy density
 
vzn
@WilliamBulmer wild guessing it could be measuring curvature/ "density" of space? have been following a somewhat vaguely related inquiry
 
@0celo7 Sure sounds like random math words to me right now. Damn mud almost wrecked the car and it took us two days to get home.
 
8:12 PM
@peterh : sounds good to me. Maybe one could divide up physics stack exchange in some similar fashion.
 
While the others involve what I think is the EM Lagrangian, and the pseudoscalar, $E \cdot B$
 
@ACuriousMind Given the flow $\varphi_t$ of a vector field $X$ on a compact manifold $M$, $\varphi_t:M\to M$ for fixed $t$ is a diffeomorphism. Is there a general formula for $(\mathrm d\varphi_t)_p(v)$ for $p\in M, v\in T_pM$?
I feel like I'm missing something super easy here.
 
@0celo7 I said keep them for tomorrow, but I'm rather sure I don't know much about flows in any state
 
@ACuriousMind I'm deep in a Ricci flow book, I don't consider this technical :P
 
@vzn So, I am not yet working with the full EM stress-energy tensor, but the Maxwell stress tensor, which merely expresses how momentum is transferred through a volume element of space
 
8:15 PM
Estimating the time dependence of $|D^m\mathrm{Riem}|^2$ is technical :P
 
The Maxwell stress tensor (named after James Clerk Maxwell) is a second rank tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impossibly difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor...
 
@0celo7 The most technical question I had to solve in the last few days was how to not fall face first into deep mud :P
 
@ACuriousMind I was tricked by my prof again into reading a super hard book
 
lol, you don't learn, do you?
Or did you finish Milnor? :P
 
In continuum mechanics, the Cauchy stress tensor σ {\displaystyle {\boldsymbol {\sigma }}\,\!} , true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components σ i j {\displaystyle \sigma _{ij}\,\!} that completely define the state of stress at a point inside a material in the deformed state, placement...
 
8:17 PM
I finished Milnor a while ago
i just accepted those two theorems involving CW complexes and the rest was easy
I didn't read the last two chapters on Bott periodicity, admittedly
But this Ricci flow book is super frustrating. I can accept taking the PDE results on faith, but not understanding the geometry is really annoying.
 
8:39 PM
Hmm, this result makes no sense now
Unless singular points of vector fields do something with the flows
@ACuriousMind If $p\in M$ is a point at which $X_p=0$, then does the associated flow satisfy $\varphi_t(p)=p$?
I guess it does
The integral curves starting at $p$ should just stay there.
Ok, the thing I'm looking at does make sense
 
@WilliamBulmer : interesting stuff William.
 
@JohnDuffield Which part?
So, I wonder if I might get my answer by the fact that that second link Provided mentions that pressure involves the trace of the cauchy stress tensor
Gahh, I feel so ignorant
hmmm...actually...so, if we're talking about the Maxwell stress tensor, then it specifically represents momentum density changes that aren't given by the Lorentz force
 
8:55 PM
@WilliamBulmer That's not some deep physics, it's just linear algebra: Being an eigenvalue of $T$ means that $\det(T - \lambda I)=0$. The Maxwell stress is (modulo constants) $T = E\otimes E+B\otimes B + \rho I$ where $\rho$ is the density. The rank of $E\otimes E$ and $B\otimes B$ is 1 (or 0), so $T-\rho I$ has maxmal rank 2, and as a 3x3 matrix it will therefore have $\rho$ as an eigenvalue.
 
@ACuriousMind And that's not technical???
 
Since the stress tensor is usually not used as acting on some vectors, there's little significance to its eigenvalues
@0celo7 No, that's linear algebra. First semester stuff
 
@ACuriousMind In GR the eigenvalues are important!
 
@0celo7 Oh?
 
Is the trace not the sum of the eigenvalues?
 
8:57 PM
(note that the stress tensor is not the full EM stress-energy tensor you might know from GR)
 
@ACuriousMind oh
@ACuriousMind It's not clear to me why the rank of $E\otimes E$ is $\le 1$.
Seems very technical.
 
@0celo7 Choose $E$ as an orthonormal basis vector, the matrix $E\otimes E$ contains one 1 and only zeros otherwise
If $E=0$, it's just zeros.
 
@ACuriousMind That would require me to know that rank is basis independent
 
@WilliamBulmer : the way the electromagnetic stress tensor resembles the continuum-mechanics Cauchy stress tensor. And the way the expression for the speed of a shear wave in a solid c = √(G/ρ) resembles c = 1/√(ε0μ0). The reciprocal is there because permittivity is a "how-easy" measure rather than a "how-hard" measure.
 
...yes, obviously.
 
8:59 PM
@ACuriousMind TIL.
 
0
Q: In what capacity are physicists still interested in classical scattering?

DiffycueI'm working through the scattering sections of Mechanics by Landau and Lifshitz, and wanted to know if/how physicists today employ the methods of purely classical scattering processes. As far as I can surmise, these are useful for Understanding Rutherford Scattering in the gold foil experiment ...

Too broad?
 
@0celo7 How do you even define rank in your world where that's not evident? The standard definition I'd give would be that it's the dimension of the image of the linear operator.
 
@Qmechanic : IMHO it's an intelligent question from the sort of person PSE should not discourage.
I have to go I'm afraid. Bye all.
 
@ACuriousMind Let $(S^2,g)$ be a gradient Ricci soliton. One can show that that soliton function $f:S^2\to \Bbb R$ satisfies $\nabla^2f=(\rho-\frac{1}{2}s)g$, where $s$ is the scalar curvature and $\rho$ is the soliton constant. Since $S^2$ is 2D, one can show (cf. Kobayashi-Nomizu Vol 2) that there is an almost complex structure $J$ and a symplectic form $\omega$ such that $g(u,Jv)=\omega(u,v)$ and $\omega(u,Jv)=g(u,v)$. Furthermore, $\nabla J=0$. Letting $\xi=\mathrm{grad}(f)$, one can [cont.]
 
@ACuriousMind I am thinking about your answer hard intensely :)
 
9:04 PM
@0celo7 However, it would indeed also follow from the tracelessness of the stress-energy that the trace of the stress tensor must be the energy density.
@0celo7 I don't know a thing about Ricci flows
 
@ACuriousMind show that $J\xi$'s flow is a 1-parameter group of isometries. Let $p\in S^2$ be a critical point of $p$. Let $a=\rho-\frac{1}{2}s(a)$. Then $D^2f=ag$ at $p$. SOMEHOW this implies $(\mathrm d\varphi_t)_p(v)=\cos(at)v+\sin(at)Jv$ for all $v\in T_pS^2$.
@ACuriousMind I don't think Ricci flow is important here
One uses Ricci solitons to construct Ricci flows, a priori they are not linked
So no PDE shenanigans yet
 
Yeah, I'm not going to read that. I asked you to not bother me with technical questions and you do the very opposite.
 
I'm typing it here for reference so when you want to read it, you can
I am also of the mind that asking questions helps me solve them
 
@ACuriousMind Ohh, haha. Duh. $\sigma+\rho I = 0$ satisfies the eigenvalue equation. Where $\sigma$ is the Maxwell stress tensor.
@ACuriousMind Still, we have the two other eigenvalues to account for, and they, too have units in energy density
@ACuriousMind But there is significance to its eigenvectors, which are the principal stresses
@ACuriousMind And surely, one could come up with vectors for this to act on, right? I mean, the canonical example would be faces that experience no "shear" transfers of momentum (I don't say forces, because the Maxwell stress tensor seems to represent a generalized momentum flow, rather than an actual force like the Lorentz force).
 
@WilliamBulmer I'm not an expert on elasicity theory, but I think the stresses are the eigenvalues, not the eigenvectors. Also, the principal stresses - like the principal axes of the moment of inertia - are nice because of computational simplicity. You can compute everything you can compute from them in any other basis, it's just going to be more tedious.
@WilliamBulmer What do you mean, "surely, one could come up with vectors for this to act on"?
I don't see any obvious candidates
 
9:18 PM
@ACuriousMind I am referring specifically to en.wikipedia.org/wiki/…
@ACuriousMind If $\sigma$ represented a "force density", which it doesn't, it would represent the forces on a volume element of material. However, it is more of a flow (as I understand it. Maybe I am wrong)
@ACuriousMind hold on...I should have finished my thought before writing that
 
Well, okay - you let it act on normal vectors to get stress vectors on the planes they're normal to. Using the eigenvectors gives you the principal stresses, which are just the same normal vector multiplied by the eigenvalue.
@WilliamBulmer Since it's the spatial part of the EM stress-energy, it represents a pressure (diagonal elements) and shear stresses (off-diagonals).
So yes, it does represent a force density - pressure is force per unit area after all.
 
@Qmechanic Yeah that's rather broad. However, I strongly discourage closing questions with the more vague close reasons (like "too broad") without an accompanying comment explaining how to fix.
 
9:49 PM
@ACuriousMind So, let's approach this from a different angle. Yes, I get that it represents pressure. On the other hand, it also has units in energy density. To me, that suggests that there is an interpretation involving energy density
 
not another dimensional analyst!
 
@ACuriousMind haha
 
Units are never evidence of anything
 
@ACuriousMind Well, it is evidence. Just not sufficient evidence
 
@ACuriousMind In his GR book, Zee obtains the Hawking temperature and Bernstein-Hawking entropy using dimensional analysis
He's only off by $8\pi$ or something
 
9:51 PM
As in, it can be used as a compass directing you to a deeper relationship
or, of course, it could be garbage as demonstrated by the XKCD cartoon
Hence, my wondering if there is physical significance to one of the eigenvalues actually being energy density, and yes, I understand your observation that it is a consequence of basic linear algebra
 
@ACuriousMind wrong units are always a pretty good sign of an error :D
 
This is probably related to this conversation: physics.stackexchange.com/questions/216342/…
 
@WilliamBulmer Well...pressure always has units of energy volume density - pressure is force per unit area, and energy is force times unit length, so pressure is energy per unit volume
 
@ACuriousMind I understand that. I am asking if there is a physical explanation for that, or is it only coincidental.
 
@WilliamBulmer Aha!
 
9:57 PM
@ACuriousMind Aha? Or Haha?
 
Aha! (because I'm beginning to understand your actual question, which seems now only loosely to any specific eigenvalues)
There are a few instances where pressure is an energy density, see KyleKanos' answer here
 
@ACuriousMind Haha, yes, I guess it is. It is related in that the trace of the stress tensor represents pressure
 
@ACuriousMind NON-TECHNICAL QUESTION:
 

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